Description
1) Convert the ones in Cartesian form to polar form and the ones in polar form to
Cartesian form.
a) 3ejπ/3+ 4e-jπ/6
b) (1-j)2
c) (√3 –j3)10
d) (√2 + j√2) / (1 + j√3)
e) Re { je-jπ/3}
f) j(1-j)
g) (√3 –j3)-1
2) Define the following complex exponential signal:
s(t) = 5ejπ/3e
j10πt
a)Make a plot of si(t) = Im{s(t)}. Pick a range of values for t that will include exactly
three periods of the signal.
b)Make a plot of q(t) = Im{ṡ(t)}, where the dot mean differentiation with respect to time
t. Again plot three cycles of the signal.
3) A signal composed of sinusoids is given by the equation
x(t) = 100cos(40πt – π / 4) + 80sin(80πt) – 60cos(120πt + π / 6)
a) Sketch the spectrum of this signal indicating the complex size of each frequency
component. You do not have to make separate plots for real/imaginary parts or
magnitude/phase. Just indicate the complex phasor value at the appropriate
frequency.
b) Is x(t) periodic? If so, what is the period?
c) Now consider a new signal y(t) = x(t)+90 cos(60πt +π/6). How is the spectrum
changed? Is y(t) periodic? If so, what is the period?
d) Finally, consider another new signal w(t) = x(t)+10 cos(280t+π/2). How is the
spectrum changed? Is w(t) periodic? If so, what is the period? If not, why not?
4) Let
x(t) = { t, 0 ≤ t ≤ 1
2 – t, 1 ≤ t ≤ 2
be a periodic signal with fundamental period T =2 and Fourier coefficients ak.
a) Determine the value of a0.
b) Determine the Fourier series representation of dx(t)/dt.
c) Use the result part (b) and differentiation property of continuous – time Fourier
series to help determine the Fourier series coefficients of x(t).
5) Suppose that a discrete- time signal x[n] is given by the formula
x[n] = 2.2 cos(0.3πn – π / 3)
and that it was obtained by sampling a continuous- time signal x(t)= Acos(2 πf0t + Ø)
at a sampling rate of fs = 6000 samples/ sec. Determine three different continuous-
time signals that could have produced x[n]. All these continuous time signals should
have a frequency less than 8kHz. Write the mathematical formula for all three.
6) Let
x[n] = δ[n] + 2 δ[n-1] – δ[n-3] and h[n]= 2 δ[n+1] + 2 δ[n-1].
Compute each of the following convolutions.
a) y1[n] = x[n] * h[n] b) y2[n] = x[n + 2] * h[n] c) y3[n] = x[n] * h[n+2]
7) A linear time-invariant system is described by the difference equation
y[n] = 2x[n] – 3x[n-1] + 2x[n-2]
a) Find the frequency response H(ejw); then express it as a mathematical formula, in
polar (magnitude and phase)
b) H(ejw) is a periodic function of w; determine the period.
c) Plot the magnitude and phase of H(ejw) as a function of w for –π<w<3π.
d) Find all frequencies w, for which the output response to the input ejwn is zero.
e) When the input to the system is x[n] = sin(πn/13), determine the output signal and
express it in form y[n] = A(cosw0n + Ø).
8) Consider a discrete time LTI system with impulse response
h[n] = { 1, 0 ≤ n ≤ 2
-1, -2 ≤ n ≤ -1
0, otherwise
Given that the input to the system is
x[n] =∑ δ[n − 4k]
∞
𝑘=−∞
determine the Fourier series coefficients of the output y[n].
9) Consider the cosine wave
x(t) = 10cos(880 πt + Ø)
Suppose that we obtain a sequence of numbers by sampling the waveform at equally
spaced time instants nTs. In this case the resulting sequence would have the values
x[n] = n(nTs) = 10 cos (880πnTs + Ø)
for −∞ < n < ∞ Suppose that Ts= 0.0001 sec.
a) How many samples will be taken in one period of the cosine wave ?
b) Now consider another wave form y(t) such that
y(t) =10 cos(w0t+ Ø)
Find a frequency w0>880π such that y(nTs) = x(nTs) for all integers n.
Hint: Use the fact that cos(θ + 2πn) = cos(θ) if n is an integer.
c) For the frequency found in (b), what is the total number of samples taken in one
period of x(t)?
10) For each of the following systems, determine whether or not the systems is linear,
time-invariant and causal.
a) y[n] = x[-n]
b) y[n] = x[n-2] – 2 x[n-8]
c) y[n] = Even{x[n-1]}
d) y[n] = nx[n]
e) y[n] = x[4n+1]
